Order = 10200960 = 27.32.5.7.11.23.
Mult = 1.
Out = 1.

Porting notes

Porting incomplete.

Standard generators

Standard generators of M23 are a, b where a has order 2, b has order 4, ab has order 23 and ababababbababbabb has order 8.

Black box algorithms

Finding generators

Group Algorithm File
M23
Download

Checking generators (semi-presentations)

Group Semi-presentation File
M23 〈〈 a, b | o(a) = 2, o(b) = 4, o(ab) = 23, o(abababab2abab2ab2) = 8 〉〉 Download

Presentations

Group Presentation Link
M23 〈 a, b | a2 = b4 = (ab)23 = (ab2)6 = [a, b]6 = (abab−1ab2)4 = (ab)3ab−1ab2(abab−1)2(ab)3(ab−1)3 = (abab2)3(ab2ab−1)2abab2abab−1ab2 = 1 〉 Details
M23 〈 a, b | a2 = b4 = (ab2)6 = (abab−1ab2)4 = abababab−1ab2abab−1abab−1abababab−1ab−1ab−1 = abab2abab2abab2ab2ab−1ab2ab−1abab2abab−1ab2 = abab2ab2abab2ab2abab2ab2abab2ab2ab−1ab2ab2ab−1ab2ab2 = ababababab2abab−1abab2ababab−1ab2abab2ab2abab−1ab−1abab2 = 1 〉 Details

Representations

Representations of M23

Maximal subgroups

Maximal subgroups of M23

Subgroup Order Index Programs/reps
M22 443 520 23Program: Standard generators
L3(4):22 40 320 253Program: Generators
24:A7 40 320 253Program: Generators
A8 20 160 506Program: Standard generators
M11 7 920 1 288Program: Standard generators
24:(3 × A5):2 5 760 1 771Program: Generators
23:11 253 40 320Program: Generators

Conjugacy classes

Conjugacy classes of M23

Conjugacy class Centraliser order Power up Class rep(s)
1A10 200 960 ababbababbababbababbababbababbababbababb
2A2 688 4A 6A 8A 14A 14B ababbababbababbababb
3A180 6A 15A 15B abbabb
4A32 8A ababbababb
5A15 15A 15B (abababb)3
6A12 abb
7A14 7B3 14A 14B (bababababbabbbababababbabb)3
7B14 7A3 14A 14B bababababbabbbababababbabb
8A8 ababb
11A11 11B2 ababababbababababb
11B11 11A2 ababababb
14A14 14B3 (bababababbabb)3
14B14 14A3 bababababbabb
15A15 15B7 abababb
15B15 15A7 (abababb)7
23A23 23B5 (ab)5
23B23 23A5 ab

Download words for class representatives.